l aware that an explanation laden to such an extent with mysteries explains but very little more than silence does； nevertheless； it is at least a silence traversed by restless murmurs； and sedulous whispers that are better than the gloomy and hopeless ignorance to which we would have perforce to resign ourselves if we did not； in spite of all； to perform the great duty of man； which is to discover a spark in the darkness。
It goes without saying that objections are raised from every side。 Among men； arithmetical prodigies are looked upon as monsters； as a sort of extremely rare teratological phenomenon。 We can count； at most； half…a…dozen in a century； whereas； among horses； the faculty would appear to be almost general； or at least quite common。 In fact； out of six or seven stallions whom Krall tried to initiate into the secrets of mathematics； he found only two that appeared to him too poorly gifted for him to waste time on their education。 These were； I believe； two thoroughbreds that were presented to him by the Grand…duke of Mecklenburg and sent back by Krall to their sumptuous stables。 In the four or five others； taken at random as circumstances supplied them； he met with aptitudes unequal； it is true； but easily developed and giving the impression that they exist normally； latent and inactive； at the bottom of every equine soul。 From the mathematical point of view； is the horse's subliminal consciousness then superior to man's？ Why not？ His whole subliminal being is probably superior to one； of greater range； younger； fresher； more alive and less heavy； since it is not incessantly attacked； coerced and humiliated by the intelligence which gnaws at it； stifles it； cloaks it and relegates it to a dark corner which neither light nor air can penetrate。 His subliminal consciousness is always present； always alert； ours is never there； is asleep at the bottom of a deserted well and needs exceptional operations； results and events before it can be drawn from its slumber and its unremembered deeps。 All this seems very extraordinary； but； in any case； we are here in the midst of the extraordinary； and this outlet is perhaps the least hazardous。 It is not a question； we must remember； of a cerebral operation； an intellectual performance； but of a gift of divination closely allied to other gifts of the same nature and the same origin which are not the peculiar attribute of man。 No observation； no experiment enables us； up to the present； to establish a difference between the subliminal of human beings and that of animals。 On the contrary； the as yet restricted number of actual cases reveals constant and striking analogies between the two。 In most of those arithmetical operations； be it noted； the subliminal of the horse behaves exactly like that of the medium in a rate of trance。 The horse readily reverses the figures of the solution； he replies； 〃37；〃 for instance； instead of 〃73；〃 which is a mediumistic phenomenon so well…known and so frequent that it has been styled 〃mirror…writing。〃 He makes mistakes fairly often in the most elementary additions； and subtractions and much less frequently in the extraction of the most complicated roots； which again； in similar cases； such as 〃xenoglossy〃 and psychometry； is one of the eccentricities of human mediumism and is explained by the same cause； namely； the inopportune intervention of the ever fallible intelligence； which； by meddling in the matter； alters the certainties of a subliminal which； when left to itself； never makes a mistake。 It is； in fact； quite probable that the horse； being really able to do the small sums； no longer relies solely on his intuition and； from that moment； gropes and flounders about。 The solution hovers between the intelligence and the subliminal and； passing from the one； which is not quite sure of it； to the other； which is not urgently appealed to； comes out of the conflict as best it may。 The case is the same with the psychometric or spiritualistic medium who seeks to profit by what he knows in the ordinary way； so as to complete the visions or revelations of his subconscious sensibility。 He； too； in this instance； is nearly always guilty of flagrant and inexplicable blunders。
Many other similarities will be found to exist； notably the way in which the lessons vary。 Nothing is more uncertain and capricious than manifestations of human mediumism。 Whether it be a question of automatic writing； psychometry； materializations or anything else； we meet with series of sittings that yield none but absurd results。 Then； suddenly； for reasons as yet obscurethe state of the weather； the presence of this or that witness； or I know not whatthe most undeniable and bewildering manifestations occur one after the other。 The case is precisely the same with the horses： their queer fancies； their unaccountable and disconcerting freaks drive poor Krall to despair。 He never opens the door of that uncertain stable； on important days； without a sinking at the heart。 Let the beard or the frown of some learned professor fail to please the horses： they will； forthwith； take an unholy delight in giving the most irrelevant answer to the most elementary question； for hours and even days on end。
Other common features are the strongly…marked personality of the mediumistic 〃raps〃 and the communications known as 〃deferred telepathic communications；〃 that is to say； those in which the answer is obtained at the end of a sitting to a question put at the beginning and forgotten by all those present。 What at first sight seems one of the strongest objections urged against the mediumism of the horse even tends to confirm it。 If the reply comes from the horse's subconsciousness； it has been asked； how is it that it should be necessary first to teach him the elements of language； mathematics and so forth； and that Berto； for instance； is incapable of solving the same problems as Mohammed？ This objection has been very ably refuted by M。 de Vesme； who writes：
〃To produce automatic writing； a medium must have learnt to write； before Victorien Sardou or Mlle Helene Schmidt could produce their mediumistic drawings and paintings； they had to possess an elementary knowledge of drawing and painting； Tartini would never have composed The Devil's Sonata in a dream； if he had not known music； and so forth。 Unconscious cerebration； however wonderful； can only take effect upon elements already acquired in some way or another。 The subconscious cerebration of a man blind from birth will not make him see colours。〃
Here； then； in this comparison which might easily be extended； are several fairly well… defined features of resemblance。 We receive a vivid impression of the same habits； the same contradictions； and the same eccentricities； and we once more recognize the strange and majestic shadow of our unknown guest。
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One great objection remains； based upon the very nature of the phenomenon； upon the really inseparable distance that separates the whole life of the horse from the abstract and impenetrable life of numbers。 How can his subliminal consciousness interest itself for a moment in signs that represent nothing to him； have no relation to his organism and will never touch his existence？ But in the first place； it is just the same with the child or the illiterate calculator。 He is not interested either in the figures which he lets loose。 He is completely ignorant of the consequences of the problems which he solves。 He juggles with digits which have hardly any more meaning to him than to the horse。 He is incapable of accounting for what he does； and his subconsciousness also acts in a sort of indifferent and remote dream。 It is true that； in his case； we can appeal to heredity and to memory； but is this difference enough to settle the difficulty and definitely to separate the two phenomena？ To appeal to heredity is still to appeal to the subliminal； and it is not at all certain that the latter is limited by the interest of the organism sheltering it。 It appears； on the contrary； in many circumstances； to spread and extend far beyond that organism in which it is domiciled； one would say； accidentally and provisionally。 It likes to show； apparently； that it is in relation with all that exists。 It declares itself； as often as possible； universal and impersonal。 It has but a very indifferent care； as we have seen in the matter of apparitions and premonitions； for the happiness and even the safety of its host and protector。 It prophesies to its companion of a lifetime events which he cannot avoid or which do not concern him。 It makes him see beforehand； for instance； all the circumstances of the death of a stranger whom he will only hear of after the event； when this event is irrevocable。 It brings a crowd of barren presentiments and conjures up veridical hallucinations that are wholly alien and idle。 With psychometric； typtological or materializing mediums； it practises art for art's sake； mocks at space and time； passes through personalities； sees through solid bodies； brings into communication thoughts and motions worlds apart； reads souls and lives by the light of a flower； a rag of a scrap of paper； and all this for nothing； to amuse itself； to astonish us； because it adores the superfluous； the incoherent； the unexpected； the improbable； the bewildering； or rather； perhaps； because it is a huge； rough； undisciplined force still struggling in the darkness and coming to the surface only by wild fits and starts； because it is an enormous expansion of a spirit striving to collect itself； to achieve consciousness； to make itself of service and to obtain a hearing。 In any case； for the time being； it appeals just what we have described； and would be unlike itself if it behaved any otherwise in the case that puzzles us。
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Lastly； to close this chapter； let us remark that it is nearly certain that the solution given by calculating children and horses is not of a mathematical nature at all。 They do not in any way consider the problem or the sum to be worked。 They simply find the answer straight away to a riddle； the guessing of which is made easy by the actual nature of figures which keep their secrets badly。 To any one in the requisite state of mind； it becomes a question of a sort of elementary charad